Max-min fuzzy bi-level programming: resource sharing system with application

Abstract

This study delves into the application of min-max fuzzy relation systems, specifically focusing on equality and inequality, in the examination of educational institution networks and their resource-sharing dynamics, akin to a P2P network structure. Expanding upon this framework, this article intricately explores a mathematical system featuring three or more terminals within an education resource system, each comprising distinct sharing capabilities. The terminals exchange information on various educational facets, encompassing variations and expenditures per share data, within the framework of a programming-type fuzzy objective function, constrained by the parameters of an educational information system. In these scenarios, the transfer of resources from one terminal to another is interlinked, introducing complexities that warrant a comprehensive examination. Notably, the consideration of greater downloading resources in specific cases is proposed for enhanced practicality. The primary objective of this article is to minimize network congestion within different educational networks, given fixed priority grades assigned to the terminals. To achieve this, the concept of a Lexicographic minimum solution is introduced to address max-min fuzzy relation inequalities, aligning with the defined objectives. The article presents a detailed and effective scenario for implementing the Lexicographic minimal solution. For validation purposes, the proposed scenario is supported by illustrative examples, offering tangible insights into its applicability and efficacy.

Keywords:

• Mathematical system
• max-min fuzzy relation inequality system
• programming for fuzzy solution
• numerical example

Mathematics Subject Classifications:

• 08A72
• 15B15
• 97H30

1. Introduction

Algebra is composed of binary operations of addition and multiplication which is mostly used for the composition operators. Such composition operators play a key role for investigating the quantitative as well as qualitative relations. However, in different fields analysis has been unstable and has some drawbacks. Therefore, the composition operators have been replaced by the fuzzy addition–multiplication composition which is quite suitable for the said binary relation. By doing this, some of the well-known mathematicians showed their interest in the max-min fuzzy algebra [1–5]. This technique and scheme have been mostly used in the field of computer engineering, their management and controlling [6–10].

Firstly, the composition operator was applied to the problem of linear equations by Sanchez [11,12] and was termed as the max-min fuzzy relation system of equations. The scenario of the of fuzzy max-min composition equation system was highly different from the classical system of equations. It has been shown that the total set of solution for the consistent max-min fuzzy system is a not convex in more cases having the condition for the minimal solution which is not unique. The complete solution is achieved by the unique maximum solution along with finite minimal solutions. Many scholars have assiduously improved the body of literature by applying the above outlined methodologies to a variety of fields and carefully investigating a broad range of issues. Their committed work enhances the academic environment and offers readers who want to learn more about their interests in-depth materials. These experts give readers a rich tapestry of insights and ideas to use in their own study by examining all kinds of problems and utilizing creative approaches. They enable fellow academics to explore new frontiers in their particular professions and make major contributions to the development of knowledge through their rigorous investigation and intellectual curiosity [13–17]. In decentralized organizations, decision-making problems are generally phrased as bi-level mathematical programming problems and modelled as Stackelberg games. We call a bi-level problem involving two or more decision makers at the lower level and one decision maker at the higher level decentralized bi-level programming challenge. The usefulness of bi-level mathematical programming issues is illustrated through real-world applications in non-cooperative scenarios [18–20].

Furthermore, Sanchez applied his idea of the fuzzy max-min relation equation in the filed of medical diagnosing [12]. The fuzzy max-min composition problem of equality and inequality has been successively used for the investigation of different types of practical fields like the fuzzy inferencing systems [21], imaging compressing and re-construction [22,23], education of engineering [24], three times media looking systems applying HTTP guards [25], peer-to-peer (P2P) systems of networking [26], bit-torrent-like peer-to-peer (BTP2P) scripts-sharing system, [27–33], goods supply [34–36] and wire-less system [37,38] and may other such as [39–42].

The addition–multiplication fuzzy composition system has been implemented in the file-sharing system for educational purposes, as detailed in [43,44]. Consequently, it is proposed to extend the application of this relational model to the educational resources information system, which comprises three key components. The objective is to analyse and optimize the interplay and transformations among these components, fostering a dynamic environment that can effectively address the challenges posed by educational competitions. To alleviate network congestion in educational resource systems, such as P2P network systems, an effective approach involves the minimization of compartmental values (${\mathbf{I}}_1$, ${\mathbf{I}}_2$, ${\mathbf{I}}_3$) within the specified system [45]. Accordingly, the objective function in this context is designed to $\min {\mathbf{I}}_1\vee {\mathbf{I}}_2\vee {\mathbf{I}}_3\vee \cdots \vee {\mathbf{I}}_n.$In the context of P2P education, these interconnected quantities – interest rate ${\mathbf{I}}_1$ on education, demand for investing in education ${\mathbf{I}}_2$ and the index of the price of education ${\mathbf{I}}_3$ serve as crucial components. In a P2P educational network, these quantities, existing as terminals, are linked through parameters facilitating the seamless transfer of knowledge resources. This intricate network extends beyond a mere three quantities, possibly encompassing a variable number of interconnected components denoted by n. The width between the agents ${\mathbf{I}}_1$, ${\mathbf{I}}_2$ and ${\mathbf{I}}_3$ is written by $s_{\mathrm{ij}}$. It means that when the jth quantity resources are transferred to the ith quantity whose level is given as $s_{\mathrm{ij}}\wedge {\mathbf{I}}_i,$where ${\mathbf{I}}_i$ shows the quality of the transformation for the three classes used the system along with description of the used symbols and parameters [45] as follows: (1) $\begin{array}{rl}\dot{{\mathbf{I}}_1}(t)& =\eta {\mathbf{I}}_3+{\mathbf{I}}_2{\mathbf{I}}_1-\mathrm{\Omega }{\mathbf{I}}_1,\\ \dot{{\mathbf{I}}_2}(t)& =-K{\mathbf{I}}_2-(\alpha ){\mathbf{I}}_1^2+\lambda ,\\ \dot{{\mathbf{I}}_3}(t)& =-\rho {\mathbf{I}}_2-\delta {\mathbf{I}}_1-ϵ{\mathbf{I}}_3,\end{array}$(1) where the rate of resources sharing by ${\mathbf{I}}_3$ to ${\mathbf{I}}_1$ is denoted by η. Ω is the rate of loss of sharing resources by ${\mathbf{I}}_1$, K is the rate of loss of resources by ${\mathbf{I}}_2$, α is the rate of sharing resources by ${\mathbf{I}}_1$ with ${\mathbf{I}}_2$, λ is the rate of sharing resources from any source to ${\mathbf{I}}_2$, rate of sharing of resources by ${\mathbf{I}}_3$ with ${\mathbf{I}}_2$ is denoted by ρ, rate of sharing of resources by ${\mathbf{I}}_3$ with ${\mathbf{I}}_1$ is denoted by δ, the rate of loss of resources from ${\mathbf{I}}_3$ is ϵ. The above equation can be written as (2) $\begin{array}{rl}{b}_1& \le \eta {\mathbf{I}}_3+{\mathbf{I}}_2{\mathbf{I}}_1-\mathrm{\Omega }{\mathbf{I}}_1\le d_1,\\ b_2& \le -K{\mathbf{I}}_2-(\alpha ){\mathbf{I}}_1^2+\lambda \le d_2,\\ b_3& \le -\rho {\mathbf{I}}_2-\delta {\mathbf{I}}_1-ϵ{\mathbf{I}}_3\le d_3.\end{array}$(2) By conversion of the max-min fuzzy composition inequality as follows: (3) $\begin{array}{rl}{b}_1& \le (r_{11}\wedge {\mathbf{I}}_1)\vee (r_{12}\wedge {\mathbf{I}}_2)\vee (r_{13}\wedge {\mathbf{I}}_3)\le d_1,\\ b_2& \le (r_{21}\wedge {\mathbf{I}}_1)\vee (r_{22}\wedge {\mathbf{I}}_2)\vee (r_{23}\wedge {\mathbf{I}}_3)\le d_2,\\ b_3& \le (r_{31}\wedge {\mathbf{I}}_1)\vee (r_{31}\wedge {\mathbf{I}}_2)\vee (r_{33}\wedge {\mathbf{I}}_3)\le d_3,\end{array}$(3) or we can extend it to an mn-array as follows as shown in Figure 1 (4) $\{\begin{array}{l}{b}_1\le (s_{11}\wedge {\mathbf{I}}_1)\vee (s_{12}\wedge {\mathbf{I}}_2)\vee (s_{13}\wedge {\mathbf{I}}_3)\vee \cdots \vee (s_{1n}\wedge {\mathbf{I}}_n)\le d_1\\ b_2\le (s_{21}\wedge {\mathbf{I}}_1)\vee (s_{22}\wedge {\mathbf{I}}_2)\vee (s_{23}\wedge {\mathbf{I}}_3)\vee \cdots \vee (s_{2n}\wedge {\mathbf{I}}_n)\le d_2\\ b_3\le (s_{31}\wedge {\mathbf{I}}_1)\vee (s_{31}\wedge {\mathbf{I}}_2)\vee (s_{33}\wedge {\mathbf{I}}_3)\vee \cdots \vee (s_{3n}\wedge {\mathbf{I}}_n)\le d_3\\ ⋮\\ b_m\le (s_{m1}\wedge {\mathbf{I}}_1)\vee (s_{3m}\wedge {\mathbf{I}}_2)\vee (s_{3m}\wedge {\mathbf{I}}_3)\vee \cdots \vee (s_{\mathrm{mn}}\wedge {\mathbf{I}}_n)\le d_m.\end{array}$(4)

Figure 1. Chart for any network having n terminal ${\mathbf{I}}_1,{\mathbf{I}}_2,{\mathbf{I}}_3\dots {\mathbf{I}}_n$.

This can be written as an min-objective and subjective function as (5) $\begin{array}{rl}& O.fminf({\mathbf{I}}_{\mathbf{1}},{\mathbf{I}}_{\mathbf{2}}\dots {\mathbf{I}}_{\mathbf{n}})={\mathbf{I}}_{\mathbf{1}}\vee {\mathbf{I}}_{\mathbf{2}}\vee \cdots \vee {\mathbf{I}}_{\mathbf{n}},\\ & s.t.\{\begin{array}{l}{b}_1\le (s_{11}\wedge {\mathbf{I}}_1)\vee (s_{12}\wedge {\mathbf{I}}_2)\vee (s_{13}\wedge {\mathbf{I}}_3)\vee \cdots \vee (s_{1n}\wedge {\mathbf{I}}_n)\le d_1\\ b_2\le (s_{21}\wedge {\mathbf{I}}_1)\vee (s_{22}\wedge {\mathbf{I}}_2)\vee (s_{23}\wedge {\mathbf{I}}_3)\vee \cdots \vee (s_{2n}\wedge {\mathbf{I}}_n)\le d_2\\ b_3\le (s_{31}\wedge {\mathbf{I}}_1)\vee (s_{31}\wedge {\mathbf{I}}_2)\vee (s_{33}\wedge {\mathbf{I}}_3)\vee \cdots \vee (s_{3n}\wedge {\mathbf{I}}_n)\le d_3\\ ⋮\\ b_m\le (s_{m1}\wedge {\mathbf{I}}_1)\vee (s_{3m}\wedge {\mathbf{I}}_2)\vee (s_{3m}\wedge {\mathbf{I}}_3)\vee \cdots \vee (s_{\mathrm{mn}}\wedge {\mathbf{I}}_n)\le d_m.\end{array}\end{array}$(5) Write the optimal solution of (3(3) $\begin{array}{rl}{b}_1& \le (r_{11}\wedge {\mathbf{I}}_1)\vee (r_{12}\wedge {\mathbf{I}}_2)\vee (r_{13}\wedge {\mathbf{I}}_3)\le d_1,\\ b_2& \le (r_{21}\wedge {\mathbf{I}}_1)\vee (r_{22}\wedge {\mathbf{I}}_2)\vee (r_{23}\wedge {\mathbf{I}}_3)\le d_2,\\ b_3& \le (r_{31}\wedge {\mathbf{I}}_1)\vee (r_{31}\wedge {\mathbf{I}}_2)\vee (r_{33}\wedge {\mathbf{I}}_3)\le d_3,\end{array}$(3) ) by ${\mathrm{\Omega }}^1$ then we can write as (6) $\{\begin{array}{l}O.fminf({\mathbf{I}}_{\mathbf{1}},{\mathbf{I}}_{\mathbf{2}}\dots {\mathbf{I}}_{\mathbf{n}})={\mathbf{I}}_{\mathbf{1}}\vee {\mathbf{I}}_{\mathbf{2}}\vee \cdots \vee {\mathbf{I}}_{\mathbf{n}},\\ s.t.\mathbf{I}\in {\mathrm{\Omega }}^1.\end{array}$(6) Choosing the different important degrees of the each compartmental terminals in a P2P network system related to any problem, this work defines the priority grade to the agents' terminals. To minimize any network congestion while taking some fixed priority grades of terminals, the idea of the lexicographic minimal solution of the max-min fuzzy relation system (3(3) $\begin{array}{rl}{b}_1& \le (r_{11}\wedge {\mathbf{I}}_1)\vee (r_{12}\wedge {\mathbf{I}}_2)\vee (r_{13}\wedge {\mathbf{I}}_3)\le d_1,\\ b_2& \le (r_{21}\wedge {\mathbf{I}}_1)\vee (r_{22}\wedge {\mathbf{I}}_2)\vee (r_{23}\wedge {\mathbf{I}}_3)\le d_2,\\ b_3& \le (r_{31}\wedge {\mathbf{I}}_1)\vee (r_{31}\wedge {\mathbf{I}}_2)\vee (r_{33}\wedge {\mathbf{I}}_3)\le d_3,\end{array}$(3) ) is introduced.

2. Preliminaries

The problem (3(3) $\begin{array}{rl}{b}_1& \le (r_{11}\wedge {\mathbf{I}}_1)\vee (r_{12}\wedge {\mathbf{I}}_2)\vee (r_{13}\wedge {\mathbf{I}}_3)\le d_1,\\ b_2& \le (r_{21}\wedge {\mathbf{I}}_1)\vee (r_{22}\wedge {\mathbf{I}}_2)\vee (r_{23}\wedge {\mathbf{I}}_3)\le d_2,\\ b_3& \le (r_{31}\wedge {\mathbf{I}}_1)\vee (r_{31}\wedge {\mathbf{I}}_2)\vee (r_{33}\wedge {\mathbf{I}}_3)\le d_3,\end{array}$(3) ) can be written in the short form as (7) $b_i^t\le (\mathbf{B}\circ {\mathbf{I}}_i^t)\le d_i^t,$(7) where $\mathbf{B}=(s_{\mathrm{ij}}{)}_{3\times 3}$, ${\mathbf{I}}_i=({\mathbf{I}}_1,{\mathbf{I}}_2,{\mathbf{I}}_3)$, $b_i=b_1,b_2,b_3$, $d_i=d_1,d_2,d_3$ and ‘°’ shows the max-min fuzzy composition.

Definition 2.1

The solution set of the system (3(3) $\begin{array}{rl}{b}_1& \le (r_{11}\wedge {\mathbf{I}}_1)\vee (r_{12}\wedge {\mathbf{I}}_2)\vee (r_{13}\wedge {\mathbf{I}}_3)\le d_1,\\ b_2& \le (r_{21}\wedge {\mathbf{I}}_1)\vee (r_{22}\wedge {\mathbf{I}}_2)\vee (r_{23}\wedge {\mathbf{I}}_3)\le d_2,\\ b_3& \le (r_{31}\wedge {\mathbf{I}}_1)\vee (r_{31}\wedge {\mathbf{I}}_2)\vee (r_{33}\wedge {\mathbf{I}}_3)\le d_3,\end{array}$(3) ) is written by $\mathbf{I}(\mathbf{B},b,d)$, i.e. $\mathbf{I}(\mathbf{B},b,d)=\{{\mathbf{I}}_i\in [0,1{]}^3|b_i^t\le \mathbf{B}\circ {\mathbf{I}}_i^t\le d_i^t\}.$Furthermore, model (3(3) $\begin{array}{rl}{b}_1& \le (r_{11}\wedge {\mathbf{I}}_1)\vee (r_{12}\wedge {\mathbf{I}}_2)\vee (r_{13}\wedge {\mathbf{I}}_3)\le d_1,\\ b_2& \le (r_{21}\wedge {\mathbf{I}}_1)\vee (r_{22}\wedge {\mathbf{I}}_2)\vee (r_{23}\wedge {\mathbf{I}}_3)\le d_2,\\ b_3& \le (r_{31}\wedge {\mathbf{I}}_1)\vee (r_{31}\wedge {\mathbf{I}}_2)\vee (r_{33}\wedge {\mathbf{I}}_3)\le d_3,\end{array}$(3) ) will be consistent if $\mathbf{I}(\mathbf{B},b,d)\ne 0$, otherwise it is inconsistent.

Definition 2.2

For system (3(3) $\begin{array}{rl}{b}_1& \le (r_{11}\wedge {\mathbf{I}}_1)\vee (r_{12}\wedge {\mathbf{I}}_2)\vee (r_{13}\wedge {\mathbf{I}}_3)\le d_1,\\ b_2& \le (r_{21}\wedge {\mathbf{I}}_1)\vee (r_{22}\wedge {\mathbf{I}}_2)\vee (r_{23}\wedge {\mathbf{I}}_3)\le d_2,\\ b_3& \le (r_{31}\wedge {\mathbf{I}}_1)\vee (r_{31}\wedge {\mathbf{I}}_2)\vee (r_{33}\wedge {\mathbf{I}}_3)\le d_3,\end{array}$(3) ) the solution $\hat{\mathbf{I}}\in \mathbf{I}(\mathbf{B},b,d)$ is called the maximal solution if $\mathbf{I}\le \hat{\mathbf{I}}$, for all $\mathbf{I}\in \mathbf{I}(\mathbf{B},b,d)$. The solution $\check{\mathbf{I}}\in \mathbf{I}(\mathbf{B},b,d)$ is called the minimum solution if $\mathbf{I}\le \check{\mathbf{I}}$ implies that $\mathbf{I}\ne \check{\mathbf{I}}$.

Now, we take for the consistency of system (3(3) $\begin{array}{rl}{b}_1& \le (r_{11}\wedge {\mathbf{I}}_1)\vee (r_{12}\wedge {\mathbf{I}}_2)\vee (r_{13}\wedge {\mathbf{I}}_3)\le d_1,\\ b_2& \le (r_{21}\wedge {\mathbf{I}}_1)\vee (r_{22}\wedge {\mathbf{I}}_2)\vee (r_{23}\wedge {\mathbf{I}}_3)\le d_2,\\ b_3& \le (r_{31}\wedge {\mathbf{I}}_1)\vee (r_{31}\wedge {\mathbf{I}}_2)\vee (r_{33}\wedge {\mathbf{I}}_3)\le d_3,\end{array}$(3) ), any vector $\check{\mathbf{I}}$ for $j\in J$ as (8) ${\mathbf{I}}_j=\{i\in \mathbf{I}|s_{\mathrm{ij}}>d_i\},$(8) and we consider the vector $\hat{\mathbf{I}}=({\hat{\mathbf{I}}}_1,{\hat{\mathbf{I}}}_2,{\hat{\mathbf{I}}}_3)$ where (9) $\hat{\mathbf{I}}=\{\begin{array}{ll}1,& {\mathbf{I}}_j=0,\\ {\displaystyle \underset{i\in {\mathbb{I}}_j}{\bigwedge }{d}_i,}& {\mathbf{I}}_j\ne 0.\end{array}$(9)

Theorem 2.3

[5–7]

System (3(3) $\begin{array}{rl}{b}_1& \le (r_{11}\wedge {\mathbf{I}}_1)\vee (r_{12}\wedge {\mathbf{I}}_2)\vee (r_{13}\wedge {\mathbf{I}}_3)\le d_1,\\ b_2& \le (r_{21}\wedge {\mathbf{I}}_1)\vee (r_{22}\wedge {\mathbf{I}}_2)\vee (r_{23}\wedge {\mathbf{I}}_3)\le d_2,\\ b_3& \le (r_{31}\wedge {\mathbf{I}}_1)\vee (r_{31}\wedge {\mathbf{I}}_2)\vee (r_{33}\wedge {\mathbf{I}}_3)\le d_3,\end{array}$(3) ) is said to be consistent if the vector $\hat{\mathbf{I}}$ is the solution of system (3(3) $\begin{array}{rl}{b}_1& \le (r_{11}\wedge {\mathbf{I}}_1)\vee (r_{12}\wedge {\mathbf{I}}_2)\vee (r_{13}\wedge {\mathbf{I}}_3)\le d_1,\\ b_2& \le (r_{21}\wedge {\mathbf{I}}_1)\vee (r_{22}\wedge {\mathbf{I}}_2)\vee (r_{23}\wedge {\mathbf{I}}_3)\le d_2,\\ b_3& \le (r_{31}\wedge {\mathbf{I}}_1)\vee (r_{31}\wedge {\mathbf{I}}_2)\vee (r_{33}\wedge {\mathbf{I}}_3)\le d_3,\end{array}$(3) ). If problem (3(3) $\begin{array}{rl}{b}_1& \le (r_{11}\wedge {\mathbf{I}}_1)\vee (r_{12}\wedge {\mathbf{I}}_2)\vee (r_{13}\wedge {\mathbf{I}}_3)\le d_1,\\ b_2& \le (r_{21}\wedge {\mathbf{I}}_1)\vee (r_{22}\wedge {\mathbf{I}}_2)\vee (r_{23}\wedge {\mathbf{I}}_3)\le d_2,\\ b_3& \le (r_{31}\wedge {\mathbf{I}}_1)\vee (r_{31}\wedge {\mathbf{I}}_2)\vee (r_{33}\wedge {\mathbf{I}}_3)\le d_3,\end{array}$(3) ) is consistent the $\hat{\mathbf{I}}$ is unique maximal solution of the considered system.

In real sense, if system (3(3) $\begin{array}{rl}{b}_1& \le (r_{11}\wedge {\mathbf{I}}_1)\vee (r_{12}\wedge {\mathbf{I}}_2)\vee (r_{13}\wedge {\mathbf{I}}_3)\le d_1,\\ b_2& \le (r_{21}\wedge {\mathbf{I}}_1)\vee (r_{22}\wedge {\mathbf{I}}_2)\vee (r_{23}\wedge {\mathbf{I}}_3)\le d_2,\\ b_3& \le (r_{31}\wedge {\mathbf{I}}_1)\vee (r_{31}\wedge {\mathbf{I}}_2)\vee (r_{33}\wedge {\mathbf{I}}_3)\le d_3,\end{array}$(3) ) is consistent, it must have a unique maximal solution and minimal solution of finite order. The complete set of solution for problem (3(3) $\begin{array}{rl}{b}_1& \le (r_{11}\wedge {\mathbf{I}}_1)\vee (r_{12}\wedge {\mathbf{I}}_2)\vee (r_{13}\wedge {\mathbf{I}}_3)\le d_1,\\ b_2& \le (r_{21}\wedge {\mathbf{I}}_1)\vee (r_{22}\wedge {\mathbf{I}}_2)\vee (r_{23}\wedge {\mathbf{I}}_3)\le d_2,\\ b_3& \le (r_{31}\wedge {\mathbf{I}}_1)\vee (r_{31}\wedge {\mathbf{I}}_2)\vee (r_{33}\wedge {\mathbf{I}}_3)\le d_3,\end{array}$(3) ) may be given in Theorem 2.4.

Theorem 2.4

[46,47]

Whenever system (3(3) $\begin{array}{rl}{b}_1& \le (r_{11}\wedge {\mathbf{I}}_1)\vee (r_{12}\wedge {\mathbf{I}}_2)\vee (r_{13}\wedge {\mathbf{I}}_3)\le d_1,\\ b_2& \le (r_{21}\wedge {\mathbf{I}}_1)\vee (r_{22}\wedge {\mathbf{I}}_2)\vee (r_{23}\wedge {\mathbf{I}}_3)\le d_2,\\ b_3& \le (r_{31}\wedge {\mathbf{I}}_1)\vee (r_{31}\wedge {\mathbf{I}}_2)\vee (r_{33}\wedge {\mathbf{I}}_3)\le d_3,\end{array}$(3) ) is consistent, then the set of solution can be written as $\mathbf{I}(\mathbf{B},b,d)=\underset{\check{\mathbf{I}}\in \check{\mathbf{X}}(\mathbf{B},b,d)}{⨆}[\check{\mathbf{I}},\hat{\mathbf{I}}],$here $\hat{\mathbf{I}}$ is the maximum unique solution and $\hat{\mathbb{I}}(B,b,d)$ is the solution set of all minimal solutions [1–4].

Definition 2.5

For any matrix of the form $[a_{\mathrm{ij}}{]}_{m\times n}$ the characteristic matrix of problem (3(3) $\begin{array}{rl}{b}_1& \le (r_{11}\wedge {\mathbf{I}}_1)\vee (r_{12}\wedge {\mathbf{I}}_2)\vee (r_{13}\wedge {\mathbf{I}}_3)\le d_1,\\ b_2& \le (r_{21}\wedge {\mathbf{I}}_1)\vee (r_{22}\wedge {\mathbf{I}}_2)\vee (r_{23}\wedge {\mathbf{I}}_3)\le d_2,\\ b_3& \le (r_{31}\wedge {\mathbf{I}}_1)\vee (r_{31}\wedge {\mathbf{I}}_2)\vee (r_{33}\wedge {\mathbf{I}}_3)\le d_3,\end{array}$(3) ) if (10) $a_{\mathrm{ij}}=\{\begin{array}{ll}1,& a_{\mathrm{ij}}\wedge {\hat{\mathbf{I}}}_j\ge b_i,\\ 0,& a_{\mathrm{ij}}\wedge {\hat{\mathbf{I}}}_j<b_i,\mathrm{\forall }i\in \mathbf{I},j\in \mathbf{J}.\end{array}$(10)

Theorem 2.6

Problem (3(3) $\begin{array}{rl}{b}_1& \le (r_{11}\wedge {\mathbf{I}}_1)\vee (r_{12}\wedge {\mathbf{I}}_2)\vee (r_{13}\wedge {\mathbf{I}}_3)\le d_1,\\ b_2& \le (r_{21}\wedge {\mathbf{I}}_1)\vee (r_{22}\wedge {\mathbf{I}}_2)\vee (r_{23}\wedge {\mathbf{I}}_3)\le d_2,\\ b_3& \le (r_{31}\wedge {\mathbf{I}}_1)\vee (r_{31}\wedge {\mathbf{I}}_2)\vee (r_{33}\wedge {\mathbf{I}}_3)\le d_3,\end{array}$(3) ) is consistent if there lies at least one non-zero element in every row of the characteristic matrix [48].

3. Method of resolution for the lexicographical minimal solution

The technique of the lexicographic minimum solution was introduced for the first time by Yang et al. [49,50] and Zhou et al. [51]. Generally, in this technique the process of minimizing the network congestion in the P2P network takes place by reducing the values of all compartment or terminal ${\mathbf{I}}_1,{\mathbf{I}}_2,\dots {\mathbf{I}}_n$, under some fixed priority grades of the terminal as follows: ${\mathbf{I}}_1\to {\mathbf{I}}_2\to {\mathbf{I}}_3\to \dots \to {\mathbf{I}}_n.$

Definition 3.1

Lexicographic Order [50]

Take $\mathbf{I}=(I_1,I_2\dots I_n)\in X$ and $\mathbf{J}=J_1,J_2\dots J_n\in X$, then $\mathbf{I}\prec \mathbf{J}$ for any number k such that ${\mathbf{I}}_j={\mathbf{J}}_j$ for $j\in (1,2\dots k-1)$ and ${\mathbf{I}}_j\prec {\mathbf{J}}_j$. Then, ≼ is called the lexicographic order

Definition 3.2

For system (3(3) $\begin{array}{rl}{b}_1& \le (r_{11}\wedge {\mathbf{I}}_1)\vee (r_{12}\wedge {\mathbf{I}}_2)\vee (r_{13}\wedge {\mathbf{I}}_3)\le d_1,\\ b_2& \le (r_{21}\wedge {\mathbf{I}}_1)\vee (r_{22}\wedge {\mathbf{I}}_2)\vee (r_{23}\wedge {\mathbf{I}}_3)\le d_2,\\ b_3& \le (r_{31}\wedge {\mathbf{I}}_1)\vee (r_{31}\wedge {\mathbf{I}}_2)\vee (r_{33}\wedge {\mathbf{I}}_3)\le d_3,\end{array}$(3) ) the solution ${\mathbf{I}}^{\ast }\in \mathbf{X}(\mathbf{B},b,d)$ is called the lexicographic minimum solution if ${\mathbf{I}}^{\ast }⪯\mathbf{I}$, satisfied for all $\mathbf{I}\in \mathbf{X}(\mathbf{B},b,d)$ and if this exists for (3(3) $\begin{array}{rl}{b}_1& \le (r_{11}\wedge {\mathbf{I}}_1)\vee (r_{12}\wedge {\mathbf{I}}_2)\vee (r_{13}\wedge {\mathbf{I}}_3)\le d_1,\\ b_2& \le (r_{21}\wedge {\mathbf{I}}_1)\vee (r_{22}\wedge {\mathbf{I}}_2)\vee (r_{23}\wedge {\mathbf{I}}_3)\le d_2,\\ b_3& \le (r_{31}\wedge {\mathbf{I}}_1)\vee (r_{31}\wedge {\mathbf{I}}_2)\vee (r_{33}\wedge {\mathbf{I}}_3)\le d_3,\end{array}$(3) ), then it will be unique.

Theorem 3.3

[48]

The lexicographic minimal solution of problem (3(3) $\begin{array}{rl}{b}_1& \le (r_{11}\wedge {\mathbf{I}}_1)\vee (r_{12}\wedge {\mathbf{I}}_2)\vee (r_{13}\wedge {\mathbf{I}}_3)\le d_1,\\ b_2& \le (r_{21}\wedge {\mathbf{I}}_1)\vee (r_{22}\wedge {\mathbf{I}}_2)\vee (r_{23}\wedge {\mathbf{I}}_3)\le d_2,\\ b_3& \le (r_{31}\wedge {\mathbf{I}}_1)\vee (r_{31}\wedge {\mathbf{I}}_2)\vee (r_{33}\wedge {\mathbf{I}}_3)\le d_3,\end{array}$(3) ) exists, iff system (3(3) $\begin{array}{rl}{b}_1& \le (r_{11}\wedge {\mathbf{I}}_1)\vee (r_{12}\wedge {\mathbf{I}}_2)\vee (r_{13}\wedge {\mathbf{I}}_3)\le d_1,\\ b_2& \le (r_{21}\wedge {\mathbf{I}}_1)\vee (r_{22}\wedge {\mathbf{I}}_2)\vee (r_{23}\wedge {\mathbf{I}}_3)\le d_2,\\ b_3& \le (r_{31}\wedge {\mathbf{I}}_1)\vee (r_{31}\wedge {\mathbf{I}}_2)\vee (r_{33}\wedge {\mathbf{I}}_3)\le d_3,\end{array}$(3) ) is consistent.

Theorem 3.4

[48]

If problem (3(3) $\begin{array}{rl}{b}_1& \le (r_{11}\wedge {\mathbf{I}}_1)\vee (r_{12}\wedge {\mathbf{I}}_2)\vee (r_{13}\wedge {\mathbf{I}}_3)\le d_1,\\ b_2& \le (r_{21}\wedge {\mathbf{I}}_1)\vee (r_{22}\wedge {\mathbf{I}}_2)\vee (r_{23}\wedge {\mathbf{I}}_3)\le d_2,\\ b_3& \le (r_{31}\wedge {\mathbf{I}}_1)\vee (r_{31}\wedge {\mathbf{I}}_2)\vee (r_{33}\wedge {\mathbf{I}}_3)\le d_3,\end{array}$(3) ) is consistent, then its unique lexicographic minimal solution is the minimum solution.

4. Resolution technique for the lexicographic minimal solution

The scheme of the lexicographic minimal solution of problem (3(3) $\begin{array}{rl}{b}_1& \le (r_{11}\wedge {\mathbf{I}}_1)\vee (r_{12}\wedge {\mathbf{I}}_2)\vee (r_{13}\wedge {\mathbf{I}}_3)\le d_1,\\ b_2& \le (r_{21}\wedge {\mathbf{I}}_1)\vee (r_{22}\wedge {\mathbf{I}}_2)\vee (r_{23}\wedge {\mathbf{I}}_3)\le d_2,\\ b_3& \le (r_{31}\wedge {\mathbf{I}}_1)\vee (r_{31}\wedge {\mathbf{I}}_2)\vee (r_{33}\wedge {\mathbf{I}}_3)\le d_3,\end{array}$(3) ), for n sub-problems are given, as under (11) $(M_1):\{\begin{array}{l}min{f}_1(\mathbf{I})={\mathbf{I}}_1,\\ \mathrm{s}.\mathrm{t}.b^t\le (\mathbf{B}\circ {\mathbf{I}}^t)\le d^t.\end{array}$(11) and for n sub-problems (Mk) (12) $(M_k):\{\begin{array}{l}min{f}_k(\mathbf{I})={\mathbf{I}}_k,k=2,3\dots n,\\ \mathrm{s}.\mathrm{t}.b^t\le (\mathbf{B}\circ {\mathbf{I}}^t)\le d^t,\\ {\mathbf{I}}_j={\mathbf{I}}_j^{\ast },j=1,2\dots k-1,\end{array}$(12) here ${\mathbf{I}}_j^{\ast }$ is the minimal objective value of each n sub-problem. The constraint in the system (12(12) $(M_k):\{\begin{array}{l}min{f}_k(\mathbf{I})={\mathbf{I}}_k,k=2,3\dots n,\\ \mathrm{s}.\mathrm{t}.b^t\le (\mathbf{B}\circ {\mathbf{I}}^t)\le d^t,\\ {\mathbf{I}}_j={\mathbf{I}}_j^{\ast },j=1,2\dots k-1,\end{array}$(12) ) is (13) $\{\begin{array}{l}{b}^t\le (\mathbf{B}\circ {\mathbf{I}}^t)\le d^t,\\ {\mathbf{I}}_j={\mathbf{I}}_j^{\ast },j=1,2\dots k-1.\end{array}$(13) Now, we will develop a resolution scheme for solution (12(12) $(M_k):\{\begin{array}{l}min{f}_k(\mathbf{I})={\mathbf{I}}_k,k=2,3\dots n,\\ \mathrm{s}.\mathrm{t}.b^t\le (\mathbf{B}\circ {\mathbf{I}}^t)\le d^t,\\ {\mathbf{I}}_j={\mathbf{I}}_j^{\ast },j=1,2\dots k-1,\end{array}$(12) ). For this we take the characteristic matrix of system (3(3) $\begin{array}{rl}{b}_1& \le (r_{11}\wedge {\mathbf{I}}_1)\vee (r_{12}\wedge {\mathbf{I}}_2)\vee (r_{13}\wedge {\mathbf{I}}_3)\le d_1,\\ b_2& \le (r_{21}\wedge {\mathbf{I}}_1)\vee (r_{22}\wedge {\mathbf{I}}_2)\vee (r_{23}\wedge {\mathbf{I}}_3)\le d_2,\\ b_3& \le (r_{31}\wedge {\mathbf{I}}_1)\vee (r_{31}\wedge {\mathbf{I}}_2)\vee (r_{33}\wedge {\mathbf{I}}_3)\le d_3,\end{array}$(3) ) which is constraint for problem (12(12) $(M_k):\{\begin{array}{l}min{f}_k(\mathbf{I})={\mathbf{I}}_k,k=2,3\dots n,\\ \mathrm{s}.\mathrm{t}.b^t\le (\mathbf{B}\circ {\mathbf{I}}^t)\le d^t,\\ {\mathbf{I}}_j={\mathbf{I}}_j^{\ast },j=1,2\dots k-1,\end{array}$(12) ) as (14) $A^k=(a_{\mathrm{ij}}^k{)}_{m\times n},$(14) where $A^k$ is the characteristic matrix whose entries may be calculated by (15) $a_{\mathrm{ij}}^k=\{\begin{array}{ll}{a}_{\mathrm{ij}}& \mathrm{if}j\ne k,\\ 0& \mathrm{if}j=k,\mathrm{\forall }i\in \mathbf{X}.\end{array}$(15) $a_{\mathrm{ij}}^k$ are the entries of the required characteristic matrix $A^k$. Next, we write $X^k=\{i\in X|a_{\mathrm{ij}}^k=0\mathrm{\forall }j\in J\}.$

Lemma 4.1

For any of the $i\in k$ (i) and (ii) hold, (16) $\begin{array}{rl}& (\mathrm{i})a_{\mathrm{ij}}^k=0\mathrm{\forall }j\in J,\\ & (\mathrm{ii})a_{\mathrm{ij}}^k=\{\begin{array}{ll}0& \mathrm{if}j\ne k,\\ 1& \mathrm{if}j=k.\end{array}\end{array}$(16)

Proof.

(i) The first result can be proved directly from (14(14) $A^k=(a_{\mathrm{ij}}^k{)}_{m\times n},$(14) ).

(ii) By Equation (15(15) $a_{\mathrm{ij}}^k=\{\begin{array}{ll}{a}_{\mathrm{ij}}& \mathrm{if}j\ne k,\\ 0& \mathrm{if}j=k,\mathrm{\forall }i\in \mathbf{X}.\end{array}$(15) ), we have $a_{\mathrm{ij}}^k=a_{\mathrm{ij}}\mathrm{if}j\ne k\mathrm{\forall }j\in k.$Taking (i) with this result $a_{\mathrm{ij}}^k=a_{\mathrm{ij}}=0\mathrm{if}j\ne k\mathrm{\forall }j\in k.$As system (2(2) $\begin{array}{rl}{b}_1& \le \eta {\mathbf{I}}_3+{\mathbf{I}}_2{\mathbf{I}}_1-\mathrm{\Omega }{\mathbf{I}}_1\le d_1,\\ b_2& \le -K{\mathbf{I}}_2-(\alpha ){\mathbf{I}}_1^2+\lambda \le d_2,\\ b_3& \le -\rho {\mathbf{I}}_2-\delta {\mathbf{I}}_1-ϵ{\mathbf{I}}_3\le d_3.\end{array}$(2) ) is consistent, then by Theorem 2.6, every row in A has at least one non-zero entry. Then by the above result $a_{\mathrm{ik}}\ne 0$ and hence $a_{\mathrm{ik}}=1.$

Lemma 4.2

If $X^k=0$ then $\underset{i\in X^k}{\bigvee }{b}_i\le {\hat{\mathbf{I}}}_k$.

Proof.

Let for any $i\in X^k$, then by Lemma 1 $c_{\mathrm{ik}}=1$, then by Definition 2.5, $a_{\mathrm{ik}}\wedge {\hat{\mathbf{I}}}_k\ge b_i.$By this, ${\hat{\mathbf{I}}}_k\ge a_{\mathrm{ik}}\wedge {\hat{\mathbf{I}}}_k\ge b_i$. Now, by any value we can write ${\hat{\mathbf{I}}}_k\ge \underset{i\in X^k}{\bigvee }{b}_i.$

Theorem 4.1

If $X^k=0$ then the minimal solution of the objective function system (12(12) $(M_k):\{\begin{array}{l}min{f}_k(\mathbf{I})={\mathbf{I}}_k,k=2,3\dots n,\\ \mathrm{s}.\mathrm{t}.b^t\le (\mathbf{B}\circ {\mathbf{I}}^t)\le d^t,\\ {\mathbf{I}}_j={\mathbf{I}}_j^{\ast },j=1,2\dots k-1,\end{array}$(12) ) is ${\mathbf{I}}^{\ast }=0$.

Proof.

For the proof we can write $\mathbf{J}=({\mathbf{J}}_1,{\mathbf{J}}_2,\dots {\mathbf{J}}_n)=({\hat{\mathbf{I}}}_1,{\hat{\mathbf{I}}}_2\dots {\hat{\mathbf{I}}}_n).$(i) $\mathbf{J}$ is the solution of system (12(12) $(M_k):\{\begin{array}{l}min{f}_k(\mathbf{I})={\mathbf{I}}_k,k=2,3\dots n,\\ \mathrm{s}.\mathrm{t}.b^t\le (\mathbf{B}\circ {\mathbf{I}}^t)\le d^t,\\ {\mathbf{I}}_j={\mathbf{I}}_j^{\ast },j=1,2\dots k-1,\end{array}$(12) ) because ${\mathbf{J}}_j\le {\hat{\mathbf{I}}}_j,\mathrm{\forall }j\in X.$Further, ${\mathbf{J}}_j\le {\hat{\mathbf{I}}}_j$ for some $j\ne k$. Choosing any $i\in X$ and $j\in Y$ then by proposition (1) [48]. $a_{\mathrm{ij}}\wedge {\mathbf{J}}_j\le a_{\mathrm{ij}}\wedge {\hat{\mathbf{I}}}_j\le d_i$If we choose any $i\in X$, as ${\mathbf{X}}^k=0$, this implies that i will not lie in ${\mathbf{X}}^k$, it follows from the set of $X^k$, there exists $j_i\in Y$ such that $a_{i{j}_i}^k\ne 0$ then $a_{i{j}_i}^k=1$. From the definition of characteristic matrix $j_i\ne k$ and $a_{i{j}_i}=a_{i{j}_i}^k=1$. Next by Definition 2.6 $a_{i{j}_i}\wedge \widehat{{\mathbf{I}}_{j_i}}\ge b_i$. It should also be noted that ${\mathbf{J}}_{j_i}={\mathbf{I}}_{j_i}$, as $j_i=k$, therefore, $a_{i{j}_i}\wedge {\mathbf{J}}_i\ge b_i.$Next by Theorem 1 of article [48] and by above discussion we can write that $\mathbf{J}\in \mathbf{X}(\mathbf{B},b,d)$ being a solution of system (3(3) $\begin{array}{rl}{b}_1& \le (r_{11}\wedge {\mathbf{I}}_1)\vee (r_{12}\wedge {\mathbf{I}}_2)\vee (r_{13}\wedge {\mathbf{I}}_3)\le d_1,\\ b_2& \le (r_{21}\wedge {\mathbf{I}}_1)\vee (r_{22}\wedge {\mathbf{I}}_2)\vee (r_{23}\wedge {\mathbf{I}}_3)\le d_2,\\ b_3& \le (r_{31}\wedge {\mathbf{I}}_1)\vee (r_{31}\wedge {\mathbf{I}}_2)\vee (r_{33}\wedge {\mathbf{I}}_3)\le d_3,\end{array}$(3) ).

(ii) The solution of the objective function problem also obeys $\mathbf{J}$ is $f_k(\mathbf{J})={\mathbf{J}}_k=0$.

(iii) Choosing any solution $\mathbf{I}=({\mathbf{I}}_1,{\mathbf{I}}_2\dots {\mathbf{I}}_n)\in \mathbf{X}(\mathbf{B},b,d)$, and this implies that $f_k(\mathbf{I})={\mathbf{I}}_k=0\ge 0.$

By (i), (ii) and (iii), $\mathbf{J}=({\hat{\mathbf{I}}}_1,{\hat{\mathbf{I}}}_2\dots {\hat{\mathbf{I}}}_n)$ is a minimal solution of system (12(12) $(M_k):\{\begin{array}{l}min{f}_k(\mathbf{I})={\mathbf{I}}_k,k=2,3\dots n,\\ \mathrm{s}.\mathrm{t}.b^t\le (\mathbf{B}\circ {\mathbf{I}}^t)\le d^t,\\ {\mathbf{I}}_j={\mathbf{I}}_j^{\ast },j=1,2\dots k-1,\end{array}$(12) ) and their values are zero, i.e. ${\mathbf{I}}_k^{\ast }=0$.

Theorem 4.2

Let $X^k\ne 0$, then the minimal solution of system (12(12) $(M_k):\{\begin{array}{l}min{f}_k(\mathbf{I})={\mathbf{I}}_k,k=2,3\dots n,\\ \mathrm{s}.\mathrm{t}.b^t\le (\mathbf{B}\circ {\mathbf{I}}^t)\le d^t,\\ {\mathbf{I}}_j={\mathbf{I}}_j^{\ast },j=1,2\dots k-1,\end{array}$(12) ) ${\mathbf{I}}_k^{\ast }=\underset{i^{\prime }\in X^k}{\bigvee }{b}_{i^{\prime }}$.

Proof.

Writing the system into the following form as $\mathbf{J}=({\mathbf{J}}_1,{\mathbf{J}}_2,\dots {\mathbf{J}}_n)=({\hat{\mathbf{I}}}_1,{\hat{\mathbf{I}}}_2\dots {\mathbf{I}}_k^{\ast }=\underset{i^{\prime }\in X^k}{\bigvee }{b}_{i^{\prime }}\dots {\hat{\mathbf{I}}}_n).$(i) $\mathbf{J}\in \mathbf{X}(\mathbf{B},b,d)$ and following Lemma 2 ${\mathbf{J}}_k=\underset{i^{\prime }\in X^k}{\bigvee }{b}_{i^{\prime }}\le {\hat{\mathbf{I}}}_k$. Therefore ${\mathbf{J}}_j\le {\hat{\mathbf{I}}}_k$ for all $j\in Y$. Next by Proposition 1 of [48] $a_{\mathrm{ij}}\wedge {\mathbf{J}}_j\le a_{\mathrm{ij}}\wedge {\hat{\mathbf{I}}}_j\le d_i\mathrm{\forall }i\in Xj\in Y.$Next, if we chose any $i\in X$ and discus the following two cases:

Case 1. If $i\in X^k$, implies from $X^k$ definition $a_{\mathrm{ij}}^k=0\mathrm{\forall }j\in Y,$and from the Characteristic Matrix definition $a_{\mathrm{ij}}=a_{\mathrm{ij}}^k=0\mathrm{\forall }j\in Yj\ne k.$As system (3(3) $\begin{array}{rl}{b}_1& \le (r_{11}\wedge {\mathbf{I}}_1)\vee (r_{12}\wedge {\mathbf{I}}_2)\vee (r_{13}\wedge {\mathbf{I}}_3)\le d_1,\\ b_2& \le (r_{21}\wedge {\mathbf{I}}_1)\vee (r_{22}\wedge {\mathbf{I}}_2)\vee (r_{23}\wedge {\mathbf{I}}_3)\le d_2,\\ b_3& \le (r_{31}\wedge {\mathbf{I}}_1)\vee (r_{31}\wedge {\mathbf{I}}_2)\vee (r_{33}\wedge {\mathbf{I}}_3)\le d_3,\end{array}$(3) ) is consistent every row in characteristic matrix has at least one non-zero entry as given in Theorem 2.6. It is given in the above equation that $a_{\mathrm{ik}}\ne 0\Rightarrow a_{\mathrm{ik}}=1$ and so by characteristic matrix we have $a_{\mathrm{ik}}\wedge b_i\ge b_i$. Therefore, $a_{\mathrm{ik}}\wedge {\mathbf{J}}_k=a_{\mathrm{ik}}\wedge (\underset{i^{\prime }\in X^k}{\bigvee }{b}_{i^{\prime }}\ge a_{\mathrm{ik}}\wedge b_i\ge b_i).$Case 2. If i not lie in $X^k$ then by $X^k$ we have $j_i\in Y$ such that $a_{i{j}_i}^k\ne 0,\Rightarrow a_{i{j}_i}^k=1$. By the Characteristic equation we have $j_i\ne k$ and $a_{i{j}_i}=a_{i{j}_i}^k=1$, then again by the characteristic matrix $a_{i{j}_i}\wedge {\hat{\mathbf{I}}}_{j_i}\ge b_i$. Further, $j_i\ne k⟹{\mathbf{J}}_{{\mathbf{j}}_{\mathbf{i}}}=\hat{{\mathbf{I}}_{{\mathbf{j}}_{\mathbf{i}}}}$, therefore we have $a_{i{j}_i}\wedge {\mathbf{J}}_{j_i}=a_{i{j}_i}\wedge {\hat{\mathbf{I}}}_{j_i}\ge b_i.$This will hold the result of $\mathbf{J}\in \mathbf{X}(\mathbf{B},b,d)$

(ii) It is proved that $f_k(\mathbf{J})={\mathbf{J}}_k=(\underset{i^{\prime }\in X^k}{\bigvee }{b}_{i^{\prime }})$.

(iii) Now we take any $\mathbf{I}=({\mathbf{I}}_1,{\mathbf{I}}_2,\dots {\mathbf{I}}_n)\in \mathbf{X}(\mathbf{B},b,d)$, then the objective function value for $\mathbf{I}$ is $f_k(\mathbf{I})={\mathbf{I}}_k$. This implies that $f_k(\mathbf{I})\ge f_k(\mathbf{J}),$ i.e. ${\mathbf{I}}_k\ge \underset{i^{\prime }\in X^k}{\bigvee }{b}_{i^{\prime }}$, by contradiction. Next suppose ${\mathbf{I}}_k<\underset{i^{\prime }\in X^k}{\bigvee }{b}_{i^{\prime }}=b_{i^{\ast }}$, where $i^{\ast }\in X^k$, implying ${\mathbf{I}}_k<b_{i^{\ast }}$, which shows that $a_{i^{\ast }k}\wedge {\mathbf{I}}_k\le {\mathbf{I}}_k<b_i^{\ast }.$By $i^{\ast }\in X^k$ follows from Lemma 1 $a_{i^{\ast }j}=0,\mathrm{\forall }j\in Y,j\ne k,$and from the characteristic matrix $a_{i^{\ast }k}\wedge {\hat{\mathbf{I}}}_k<b_i^{\ast },\mathrm{\forall }j\in Y,j\ne k.$Notice that $\mathbf{I}\in \mathbf{X}(\mathbf{X},b,d)$ is a solution of (3(3) $\begin{array}{rl}{b}_1& \le (r_{11}\wedge {\mathbf{I}}_1)\vee (r_{12}\wedge {\mathbf{I}}_2)\vee (r_{13}\wedge {\mathbf{I}}_3)\le d_1,\\ b_2& \le (r_{21}\wedge {\mathbf{I}}_1)\vee (r_{22}\wedge {\mathbf{I}}_2)\vee (r_{23}\wedge {\mathbf{I}}_3)\le d_2,\\ b_3& \le (r_{31}\wedge {\mathbf{I}}_1)\vee (r_{31}\wedge {\mathbf{I}}_2)\vee (r_{33}\wedge {\mathbf{I}}_3)\le d_3,\end{array}$(3) ), following Theorem 2.6 $\mathbf{I}\le \hat{\mathbf{I}}⟹{\mathbf{I}}_j\le {\hat{\mathbf{I}}}_j\mathrm{\forall }j\in Y$, therefore, $a_{i^{\ast }j}\wedge {\mathbf{I}}_j\le a_{i^{\ast }j}\wedge {\mathbf{I}}_j<b_i^{\ast }\mathrm{\forall }j\in Y,j\ne k.$From previous two inequalities $a_{i^{\ast }j}\wedge {\hat{\mathbf{I}}}_j<b_i^{\ast },\mathrm{\forall }j\in Y.$Then by Theorem 1 [48], $\mathbf{I}$ is not solution of (3(3) $\begin{array}{rl}{b}_1& \le (r_{11}\wedge {\mathbf{I}}_1)\vee (r_{12}\wedge {\mathbf{I}}_2)\vee (r_{13}\wedge {\mathbf{I}}_3)\le d_1,\\ b_2& \le (r_{21}\wedge {\mathbf{I}}_1)\vee (r_{22}\wedge {\mathbf{I}}_2)\vee (r_{23}\wedge {\mathbf{I}}_3)\le d_2,\\ b_3& \le (r_{31}\wedge {\mathbf{I}}_1)\vee (r_{31}\wedge {\mathbf{I}}_2)\vee (r_{33}\wedge {\mathbf{I}}_3)\le d_3,\end{array}$(3) ) which is a contradiction. So by (i), (ii) and (iii) it is shown that $\mathbf{J}=({\hat{\mathbf{I}}}_1),{\hat{\mathbf{I}}}_2\dots \underset{i^{\prime }\in X^k}{\bigvee }{b}_{i^{\prime }}\dots {\hat{\mathbf{I}}}_n$ is a minimal solution (12(12) $(M_k):\{\begin{array}{l}min{f}_k(\mathbf{I})={\mathbf{I}}_k,k=2,3\dots n,\\ \mathrm{s}.\mathrm{t}.b^t\le (\mathbf{B}\circ {\mathbf{I}}^t)\le d^t,\\ {\mathbf{I}}_j={\mathbf{I}}_j^{\ast },j=1,2\dots k-1,\end{array}$(12) ) having value ${\mathbf{I}}_k^{\ast }=f_k(\mathbf{J})=\underset{i^{\prime }\in X^k}{\bigvee }{b}_{i^{\prime }}$.

5. Examples

In this section we provide some numerical examples of the considered mathematical model as a constraint max-min fuzzy formate having some objective functions.

Example 5.1

Let we have the following optimal problem: (17) $\{\begin{array}{l}min{f}_2(\mathbf{I})={\mathbf{I}}_2\\ \mathrm{s}.\mathrm{t}.b^t\le (\mathbf{B}\circ {\mathbf{I}}^t)\le d^t.\end{array}$(17) where $\begin{array}{rl}b& =(0.6,0.4,0.5),\\ d& =(0.8,0.9,0.7),\\ \mathbf{I}& =({\mathbf{I}}_1,{\mathbf{I}}_2,{\mathbf{I}}_3),\\ \mathbf{B}& =\left(\begin{array}{ccc}0.6& 0.7& 0.9\\ 0.8& 0.6& 1.0\\ 0.4& 0.8& 0.7\end{array}\right).\end{array}$For treating this problem, five steps are used:

1. First, we have to find the maximum value, i.e. $\hat{\mathbf{I}}=({\hat{\mathbf{I}}}_1,{\hat{\mathbf{I}}}_2{\hat{\mathbf{I}}}_3)$ using (18) $a_{\mathrm{ij}}\ast d_i\{\begin{array}{ll}1& \mathrm{if}{a}_{\mathrm{ij}}\le d_i,\\ d_i& \mathrm{if}{a}_{\mathrm{ij}}\ge d_i,i\in X,j\in Y,\end{array}$(18) and ${\hat{\mathbf{I}}}_j=\underset{i\in X}{\bigwedge }(a_{\mathrm{ij}}\ast d_i).$The following relation is obtained: $\begin{array}{rl}& \{\begin{array}{l}{a}_{11}\ast d_1=1,a_{12}\ast d_1=1,a_{13}\ast d_1=0.8,\\ a_{21}\ast d_2=1,a_{22}\ast d_2=1,a_{23}\ast d_2=1,\\ a_{31}\ast d_3=1,a_{32}\ast d_3=0.7,a_{33}\ast d_3=1,\end{array}\\ & \{\begin{array}{l}{\hat{\mathbf{I}}}_1=a_{11}\ast d_1\wedge a_{21}\ast d_2\wedge a_{31}\ast d_3=1\wedge 1\wedge 1=1,\\ {\hat{\mathbf{I}}}_2=a_{12}\ast d_1\wedge a_{22}\ast d_2\wedge a_{32}\ast d_3=1\wedge 1\wedge 0.7=0.7,\\ {\hat{\mathbf{I}}}_3=a_{13}\ast d_1\wedge a_{23}\ast d_2\wedge a_{33}\ast d_3=0.8\wedge 1\wedge 1=\mathrm{0.8.}\end{array}\end{array}$Shortly the following can be written as (19) $\hat{\mathbf{I}}=(1,0.7,0.8).$(19)

2. Next we use Definition 2.5 to get the Characteristic matrix as follows: ${\mathbf{C}}_{\mathrm{ij}}=\left(\begin{array}{ccc}1& 0& 1\\ 1& 1& 1\\ 0& 1& 1\end{array}\right).$

3. Using Lemma 4.1 chose k = 2, find matrix $a_{\mathrm{ij}}^2={\mathbf{C}}^2$ as under ${\mathbf{C}}^2=\left(\begin{array}{ccc}1& 0& 1\\ 1& 0& 1\\ 0& 0& 1\end{array}\right).$

4. As every row in the ${\mathbf{C}}^2$ matrix has at least one non-zero entry therefore by $X^k$ we have $X^k=0$ or $X^k$ is empty.

5. As $X^k=0$, therefore, by Theorem 4.1 the minimal value of system (17(17) $\{\begin{array}{l}min{f}_2(\mathbf{I})={\mathbf{I}}_2\\ \mathrm{s}.\mathrm{t}.b^t\le (\mathbf{B}\circ {\mathbf{I}}^t)\le d^t.\end{array}$(17) ) is ${\mathbf{I}}_2^{\ast }=0$. The maximal and minimal solutions are shown in Figure 2

Figure 2. Minimal and maximal values of ${\mathbf{I}}_1,{\mathbf{I}}_2\mathrm{and}{\mathbf{I}}_3$.

Example 5.2

Let we have the second optimal problem as under (20) $\{\begin{array}{l}min{f}_3(\mathbf{I})={\mathbf{I}}_3\\ \mathrm{s}.\mathrm{t}.b^t\le (\mathbf{B}\circ {\mathbf{I}}^t)\le d^t.\end{array}$(20) where $\begin{array}{rl}b& =(0.4,0.3,0.6),\\ d& =(0.5,0.4,0.6),\\ \mathbf{I}& =({\mathbf{I}}_1,{\mathbf{I}}_2,{\mathbf{I}}_3),\\ \mathbf{B}& =\left(\begin{array}{ccc}0.6& 0.3& 0.3\\ 0.4& 0.6& 0.3\\ 0.4& 0.5& 0.5\end{array}\right).\end{array}$For proceeding with this optimal problem, we use five steps

1. Firstly, find the maximum value, i.e. $\hat{\mathbf{I}}=({\hat{\mathbf{I}}}_1,{\hat{\mathbf{I}}}_2{\hat{\mathbf{I}}}_3)$ using (21) $a_{\mathrm{ij}}\ast d_i\{\begin{array}{ll}1& \mathrm{if}{a}_{\mathrm{ij}}\le d_i,\\ d_i& \mathrm{if}{a}_{\mathrm{ij}}\ge d_i,i\in X,j\in Y,\end{array}$(21) and ${\hat{\mathbf{I}}}_j=\underset{i\in X}{\bigwedge }(a_{\mathrm{ij}}\ast d_i),$obtain the following: $\begin{array}{rl}& \{\begin{array}{l}{a}_{11}\ast d_1=0.5,a_{12}\ast d_1=1,a_{13}\ast d_1=1,\\ a_{21}\ast d_2=1,a_{22}\ast d_2=0.4,a_{23}\ast d_2=1,\\ a_{31}\ast d_3=1,a_{32}\ast d_3=1,a_{33}\ast d_3=1,\end{array}\\ & \{\begin{array}{l}{\hat{\mathbf{I}}}_1=a_{11}\ast d_1\wedge a_{21}\ast d_2\wedge a_{31}\ast d_3=0.5\wedge 1\wedge 1=0.5,\\ {\hat{\mathbf{I}}}_2=a_{12}\ast d_1\wedge a_{22}\ast d_2\wedge a_{32}\ast d_3=1\wedge 0.4\wedge 1=0.4,\\ {\hat{\mathbf{I}}}_1=a_{13}\ast d_1\wedge a_{23}\ast d_2\wedge a_{33}\ast d_3=1\wedge 1\wedge 1=1.\end{array}\end{array}$Shortly, write the following: (22) $\hat{\mathbf{I}}=(0.5,0.4,1).$(22)

2. Next using Definition 2.5 to get the Characteristic matrix as follows: ${\mathbf{C}}_{\mathrm{ij}}=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right).$

3. Using Lemma 4.1 chose k = 3, find matrix $a_{\mathrm{ij}}^3={\mathbf{C}}^3$ as under ${\mathbf{C}}^3=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right).$

4. As the 3rd row in ${\mathbf{C}}^3$ matrix has all zero entries, therefore, by $X^k$ we have $X^k=3$.

5. As $X^k=3$, therefore, by Theorem 4.2 the minimal value of system (20(20) $\{\begin{array}{l}min{f}_3(\mathbf{I})={\mathbf{I}}_3\\ \mathrm{s}.\mathrm{t}.b^t\le (\mathbf{B}\circ {\mathbf{I}}^t)\le d^t.\end{array}$(20) ) is ${\mathbf{I}}_3^{\ast }=\underset{i\in X^3}{\bigvee }(b_i)=b_3=\mathrm{0.4.}$The maximal and minimal solutions are shown in Figure 3

Figure 3. Minimal and maximal values of ${\mathbf{I}}_1,{\mathbf{I}}_2,\mathrm{and}{\mathbf{I}}_3$.

Example 5.3

Let we have the third optimal problem as under (23) $\{\begin{array}{l}min{f}_3(\mathbf{I})={\mathbf{I}}_1\\ \mathrm{s}.\mathrm{t}.b^t\le (\mathbf{B}\circ {\mathbf{I}}^t)\le d^t.\end{array}$(23) where $\begin{array}{rl}b& =(0.4,0.3,0.6),\\ d& =(0.5,0.4,0.6),\\ \mathbf{I}& =({\mathbf{I}}_1,{\mathbf{I}}_2,{\mathbf{I}}_3),\\ \mathbf{B}& =\left(\begin{array}{ccc}0.6& 0.3& 0.3\\ 0.4& 0.6& 0.3\\ 0.4& 0.5& 0.5\end{array}\right).\end{array}$For proceeding this optimal problem, we use five steps

1. We find the maximum values as before (24) $\hat{\mathbf{I}}=(0.5,0.4,1).$(24)

2. Next using Definition 2.5 to get the Characteristic matrix as follows: ${\mathbf{C}}_{\mathrm{ij}}=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right).$

3. Using Lemma 4.1 chose k = 1, find matrix $a_{\mathrm{ij}}^1={\mathbf{C}}^1$ as under ${\mathbf{C}}^3=\left(\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right).$

4. As the 1st row in ${\mathbf{C}}^1$ matrix has all zero entries, therefore, by $X^k$ we have $X^k=1$.

5. As $X^k=1$, therefore, by Theorem 4.2 the minimal value of system (23(23) $\{\begin{array}{l}min{f}_3(\mathbf{I})={\mathbf{I}}_1\\ \mathrm{s}.\mathrm{t}.b^t\le (\mathbf{B}\circ {\mathbf{I}}^t)\le d^t.\end{array}$(23) ) is ${\mathbf{I}}_1^{\ast }=\underset{i\in X^1}{\bigvee }(b_i)=b_1=\mathrm{0.4.}$

Next the sub-problem is (25) $\{\begin{array}{l}min{f}_3(\mathbf{I})={\mathbf{I}}_2\\ \mathrm{s}.\mathrm{t}.b^t\le (\mathbf{B}\circ {\mathbf{I}}^t)\le d^t.\end{array}$(25) where $\begin{array}{rl}b& =(0.4,0.3,0.6),\\ d& =(0.5,0.4,0.6),\\ \mathbf{I}& =({\mathbf{I}}_1,{\mathbf{I}}_2,{\mathbf{I}}_3),\\ \mathbf{B}& =\left(\begin{array}{ccc}0.6& 0.3& 0.3\\ 0.4& 0.6& 0.3\\ 0.4& 0.5& 0.5\end{array}\right).\end{array}$For proceeding this optimal problem, we use five steps

1. We find the maximum values as before (26) $\hat{\mathbf{I}}=(0.5,0.4,1).$(26)

2. Next using Definition 2.5 to get the Characteristic matrix as follows: ${\mathbf{C}}_{\mathrm{ij}}=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right).$

3. Using Lemma 4.1 chose k = 2, find matrix $a_{\mathrm{ij}}^2={\mathbf{C}}^2$ as under ${\mathbf{C}}^3=\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right).$

4. As the 2nd row in ${\mathbf{C}}^3$ matrix has all zero entries, therefore, by $X^k$ we have $X^k=2$.

5. As $X^k=2$, therefore, by Theorem 4.2 the minimal value of system (25(25) $\{\begin{array}{l}min{f}_3(\mathbf{I})={\mathbf{I}}_2\\ \mathrm{s}.\mathrm{t}.b^t\le (\mathbf{B}\circ {\mathbf{I}}^t)\le d^t.\end{array}$(25) ) is ${\mathbf{I}}_2^{\ast }=\underset{i\in X^2}{\bigvee }(b_i)=b_2=\mathrm{0.3.}$

Next the sub-problem is (27) $\{\begin{array}{l}min{f}_3(\mathbf{I})={\mathbf{I}}_3\\ \mathrm{s}.\mathrm{t}.b^t\le (\mathbf{B}\circ {\mathbf{I}}^t)\le d^t.\end{array}$(27) Whose value is found in Example 5.2. So the total minimal solution in this format is ${\mathbf{I}}^{\mathbf{\ast }}=({\mathbf{I}}_1^{\ast },{\mathbf{I}}_2^{\ast },{\mathbf{I}}_3^{\ast })=(0.4,0.3,0.4).$

Example 5.4

Let we have the following optimal problem: (28) $\{\begin{array}{l}min{f}_4(\mathbf{I})={\mathbf{I}}_1\\ \mathrm{s}.\mathrm{t}.b^t\le (\mathbf{B}\circ {\mathbf{I}}^t)\le d^t.\end{array}$(28) where $\begin{array}{rl}b& =(0.36,0.45,0.42,0.37,0.31,0.40),\\ d& =(0.9,1.0,0.8,1.0,1.0,0.9),\\ \mathbf{I}& =({\mathbf{I}}_1,{\mathbf{I}}_2,{\mathbf{I}}_3,{\mathbf{I}}_4,{\mathbf{I}}_5,{\mathbf{I}}_6),\\ \mathbf{B}& =\left(\begin{array}{cccccc}0.4& 0.5& 0.44& 0.50& 0.66& 0.48\\ 0.6& 0.75& 0.74& 0.50& 0.44& 0.40\\ 0.4& 0.5& 0.7& 0.62& 0.75& 0.6\\ 0.7& 0.5& 0.44& 0.50& 0.66& 0.48\\ 0.6& 0.55& 0.74& 0.50& 0.44& 0.40\\ 0.77& 0.5& 0.3& 0.62& 0.75& 0.6\end{array}\right).\end{array}$For treating this problem, we use the five steps:

1. First, we have to find the maximum value, i.e. $\hat{\mathbf{I}}=({\hat{\mathbf{I}}}_1,{\hat{\mathbf{I}}}_2{\hat{\mathbf{I}}}_3,{\hat{\mathbf{I}}}_4,{\hat{\mathbf{I}}}_5{\hat{\mathbf{I}}}_6)$ using (29) $a_{\mathrm{ij}}\ast d_i\{\begin{array}{ll}1& \mathrm{if}{a}_{\mathrm{ij}}\le d_i,\\ d_i& \mathrm{if}{a}_{\mathrm{ij}}\ge d_i,i\in X,j\in Y,\end{array}$(29) and ${\hat{\mathbf{I}}}_j=\underset{i\in X}{\bigwedge }(a_{\mathrm{ij}}\ast d_i).$The following relation is obtained: $\begin{array}{rl}& \{\begin{array}{l}{a}_{11}\ast d_1=0.9,a_{12}\ast d_1=1,a_{13}\ast d_1=0.8,a_{14}\ast d_1=0.8,\\ a_{15}\ast d_1=0.9,a_{16}\ast d_1=0.7,\\ a_{21}\ast d_2=1,a_{22}\ast d_2=1,a_{23}\ast d_2=1,a_{24}\ast d_2=0.8,\\ a_{25}\ast d_2=1,a_{26}\ast d_2=0.9,\\ a_{31}\ast d_3=1,a_{32}\ast d_3=0.7,a_{33}\ast d_3=1,a_{34}\ast d_3=0.8,\\ a_{35}\ast d_3=0.8,a_{36}\ast d_3=0.8,\\ a_{41}\ast d_4=0.8,a_{42}\ast d_4=1,a_{43}\ast d_4=0.8,a_{44}\ast d_4=0.9,\\ a_{45}\ast d_4=0.8,a_{46}\ast d_4=1,\\ a_{51}\ast d_5=0.9,a_{52}\ast d_5=,a_{53}\ast d_5=1,a_{54}\ast d_5=0.8,\\ a_{55}\ast d_5=1,a_{56}\ast d_5=0.9,\\ a_{61}\ast d_6=1,a_{62}\ast d_6=0.8,a_{63}\ast d_6=0.9,a_{64}\ast d_6=1,\\ a_{65}\ast d_6=0.9,a_{66}\ast d_6=0.7,\end{array}\\ & \{\begin{array}{l}{\hat{\mathbf{I}}}_1=a_{11}\ast d_1\wedge a_{21}\ast d_2\wedge a_{31}\ast d_3\wedge a_{41}\ast d_4\wedge a_{51}\ast d_5\wedge a_{61}\ast d_6\\ =1\wedge 1\wedge 1\wedge 1\wedge 1\wedge 1=1,\\ {\hat{\mathbf{I}}}_2=a_{12}\ast d_1\wedge a_{22}\ast d_2\wedge a_{32}\ast d_3\wedge a_{42}\ast d_4\wedge a_{52}\ast d_5\wedge a_{62}\ast d_6\\ =1\wedge 1\wedge 0.8\wedge 1\wedge 0.9\wedge 0.8=0.8,\\ {\hat{\mathbf{I}}}_3=a_{13}\ast d_1\wedge a_{23}\ast d_2\wedge a_{33}\ast d_3\wedge a_{43}\ast d_4\wedge a_{53}\ast d_5\wedge a_{63}\ast d_6\\ =0.7\wedge 1\wedge 1\wedge 0.7\wedge 0.9\wedge 1=0.7\\ {\hat{\mathbf{I}}}_4=a_{14}\ast d_1\wedge a_{24}\ast d_2\wedge a_{34}\ast d_3\wedge a_{44}\ast d_4\wedge a_{54}\ast d_5\wedge a_{64}\ast d_6\\ =1\wedge 1\wedge 1\wedge 1\wedge 0.9\wedge 1=0.9,\\ {\hat{\mathbf{I}}}_5=a_{15}\ast d_1\wedge a_{25}\ast d_2\wedge a_{35}\ast d_3\wedge a_{45}\ast d_4\wedge a_{55}\ast d_5\wedge a_{66}\ast d_6\\ =1\wedge 1\wedge 0.8\wedge 1\wedge 0.9\wedge 0.8=0.8,\\ {\hat{\mathbf{I}}}_6=a_{16}\ast d_1\wedge a_{26}\ast d_2\wedge a_{36}\ast d_3\wedge a_{46}\ast d_4\wedge a_{56}\ast d_5\wedge a_{66}\ast d_6\\ =0.8\wedge 0.9\wedge 1\wedge 0.8\wedge 1\wedge 1=\mathrm{0.8.}\end{array}\end{array}$Shortly the following can be written as (30) $\hat{\mathbf{I}}=(1,0.8,0.7,0.9,0.8,0.8).$(30)

2. Next we use Definition 2.5 to get the Characteristic matrix as follows: ${\mathbf{C}}_{\mathrm{ij}}=\left(\begin{array}{cccccc}1& 0& 1& 1& 0& 1\\ 1& 1& 1& 1& 1& 1\\ 0& 1& 1& 0& 1& 1\\ 1& 0& 1& 1& 0& 1\\ 1& 1& 1& 1& 1& 1\\ 0& 1& 1& 0& 1& 1\end{array}\right).$

3. Using Lemma 4.1 chose k = 1, find matrix $a_{\mathrm{ij}}^1={\mathbf{C}}^1$ as under ${\mathbf{C}}^1=\left(\begin{array}{cccccc}1& 0& 1& 0& 0& 1\\ 1& 1& 1& 0& 1& 1\\ 0& 1& 1& 0& 1& 1\\ 1& 0& 1& 0& 0& 1\\ 1& 1& 1& 0& 1& 1\\ 0& 1& 1& 0& 1& 1\end{array}\right).$

4. As every row in ${\mathbf{C}}^1$ matrix has at least one non-zero entry therefore by $X^k$ we have $X^k=0$ or $X^k$ is empty.

5. As $X^k=0$, therefore, by Theorem 4.1 the minimal value of system (29(29) $a_{\mathrm{ij}}\ast d_i\{\begin{array}{ll}1& \mathrm{if}{a}_{\mathrm{ij}}\le d_i,\\ d_i& \mathrm{if}{a}_{\mathrm{ij}}\ge d_i,i\in X,j\in Y,\end{array}$(29) ) is ${\mathbf{I}}_1^{\ast }=0$.

6. Conclusion

In conclusion, this article meticulously explores the intricacies of the max-min fuzzy optimal solution, presenting it within the framework of an objective function accompanied by constraints in a mathematical model tailored specifically for the dynamic of educational resources. Through the innovative bi-programming approach applied to the three pivotal quantities within the informative system, a robust educational network system emerges. The devised procedural methodology for achieving the optimal solution takes the form of a lexicographic minimal solution, cleverly employing min-product fuzzy relations. The significance of this analysis becomes evident when applied to systems grappling with the complex interplay of education quality, investment demands and fluctuations in educational resources. The focal point shifts towards the minimal objective function with constraints, and the author introduces a technique that efficiently optimizes the solution across all three compartments. This technique, articulated through a series of theorems and lemmas, provides a systematic approach to computing the optimal solution within the considered system. To enhance practical understanding and facilitate broader application, the article offers a detailed illustration of the procedural steps through numerical examples involving each of the three compartments. This adaptability extends the study's relevance beyond its immediate scope, positioning it as a valuable resource for real-life scenarios where objective functions and subjective constraints coalesce. The insights garnered from this research thus contribute significantly to the broader landscape of problem-solving methodologies, offering nuanced perspectives for a diverse range of challenges. By minimizing compartmental values and optimizing their transformations, particularly in the P2P network, we aim to mitigate network congestion and enhance the overall efficiency of educational knowledge dissemination within the interconnected components. This approach not only underscores the importance of each individual quantity but also emphasizes the collaborative and interdependent nature of P2P education systems.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The data sets generated during and/or analysed during the current study are available from the author on reasonable request.

Additional information

Funding

This work received funding from the Conventional Project for 2022 in the ‘14th Five Year Plan’ of Philosophy and Social Sciences in Guangdong Province (GD22CJY24), 2022 and Guangdong Province Key Construction Discipline Scientific Research Capacity Improvement Project (2022ZDJS062) and Guangdong Provincial Education Science Planning Project (2021GXJK183, 2022GXJK021,2022GXJK260,2022GXJK264).